UV maping a sphere to have stripes runing between both poles?
If you do it backwards, so instead of unwrapping an icosphere you create a plane and turn it into a sphere, you have a lot more control on the results.
To turn a plane into a sphere see this answer

Unwrapping a plane will give you perfect UVs to do what you want:



.blend file

related:

What is the best way to unwrap a sphere?
https://blender.stackexchange.com/a/13927/1853